This 80 page Class Notes was uploaded by Miss Noel Mertz on Monday October 26, 2015.
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use substitution 7414 24 If all m equations in n variabl y 2 are linear then solve using ele mentary row operations ERG s ie 7381 a write the augmented matrix b use ERO s to write the augmented matrix in echelon form c rewrite the sv stem of equations using the variables 4 and use back substitution There are three possibilities i no solutions ii unique solution iii in nitely many solutions Matrices Addition subtraction multiplication of matrices Make sure you know the con ditions on the dimension of the matrices for these operations to be well de ned 82 We can only 39dividequot by a matrix if it has an inverse To nd out whether an n x n matrix A has an inverse and to compute the inverse you should 843 4 augment A by the n x n identity matrix In N use EROquots to write the matrix AIn in row reduced echelon form 63 if the resulting matrix is of the form I A then you ve found the inverse Otherwise A has no inverse An application but by no means the only reason for introducing inverses is to solve systems of n linear equations in n unknowns Write the system as AX B and trv to compute A as above The tem has a unique solution 1 1 if and only if A exists and the solution is X A Partial fractions To write the partial fraction decomposition for a rational function 27 1 if degree numerator gt degree denomenator use long division giving a poly nomial plus a rational function f x gx with degree f lt degree 94 2 factor 917 into linear and quadratic terms no complex numbers 34 for every linear term Ax8 include Mir 3 if there is a quadratic term Cu2 D41 E include 011qu 4 If there is a linear term squared Ax By include Kiwi ikuim 44 add the terms from 3 to give something like em quotI quotI quotI 3a l 5172 4a 3 20 a b 1 gm Set this equal to f x gx giving you a system of equations 3a b r coe icient of x2 in f 4a 3 20 coe i ent of a in f a b c constant term from f Solve the system Six steps to drawing graphs of rational functions Given a rational function rx f x gx with integer coe icients draw 7 2 Mar as follows 1 Is Mar common function shifted re ected stretched 2 Plot y intercepts 34 a Plot XiiIltCI39CCpts i4e nd zeros of f x by factoring b Plot vertical asymptotes ie nd zeros of gx 26 44 Look at behaviour to far right and far left ie nd the horizontal slant asymptotes 2 6 a if degree f lt degree 9 then 7 0 is asymptote A 3 if degree f 2 degree g then y LnE is asymptote ie leading coe of f divided by leading coe o g A n if degree f gt degree g then use long division to write 39rx qx rmnwindzi39r xrgx so that q x is a 39slantquot asymptote Use cunning and plot at least one point between and one point beyond all zeros and vertical asymptotes 64 Join dots with smooth curve 2 gym a L 9 r k v N MW NQQ w Lava b 1M Avw zav 7 a W Ed39g 544 p warm 31 5 Lx was a m 39 Maya 53 wws a 339 8 gt43 i A L k gi Qg Reg WWz M wig ag 7 99 3f bm g V a Vina 3a m3 gag 6L Elf Aj rxk 63L NVA gs w A b aw3 if bx V 1mg 4 S J 5x2 Eaka 5 ti Wigquot 663 I Saw W a Xx 7 3 5X49 7 5 isxni gt e 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Pg A5e Lagsir 4 M 32 futif turf E xa v kgzm walk 4sz W W way Math1010 Equation Solving Strategies 1 Linear Equations Simplify both sides ofthe equation isolate the variable 3x7215xi7 3X 61 5X 7 3X 5 5X7 52X7 22 1x Example 2 Quadratic Equations a If it has an X term in it then move everything on one side of the equation with the other side zero Try to factor it 1 If it factors set each factor equal to zero and solve 2 If it doesn39t factor use the quadratic formula ie For ax2bxc0 x ibiv sb274ac 2a b If it doesn39t have an X term in it then isolate the x2 and take the square root of both sides Rememberto put the plus or minus sign on the answer or you can complete the square Examples 3x213x10 3x213X7100 3X72x50 a 1 3X 20 or x50 3X2 or x5 x2 or x5 3 2x2x750 a2b1c5 x ilivl174275 a 2 29 x71v140 4 x1i 4 2x27500 2x250 b x225 x5 Note For quadratic equations the quadratic formula method will ALWAYS work So ifyou39d like you can always use that method 3 HigherOrder Polynomial Equations Get everything on one side of the equation with the other side zero Factor and then set each factor equal to zero and solve This is basically a more intense version ofthe 2a1 type 10x45x315x2 10x475x3715x20 5x22x27x730 Example 5x22xi3x10 5x20 or 2X 30 or x10 x0 or x2 or x1 2 4 Rational Equations Find the least common denominator LCD Multiply both sides ofthe equation by the LCD which will get rid ofthe fractions Then it turns into either a linear or quadratic equation and you solve those using the previously stated methods 7 2 7x2 x1 x57x1 LCD x1x5 x1x5 2x1x57x2x1x5 x1 x5 x1 x572x1x2x5 x572X72x27X10 7x3x27x10 0x28x7 0x7x1 0x7 or 0x1 x7 or x1 Example Note You MUST check your answers for rational equations to make sure they don39t give you a zero in the denominator ofthe original equation lfa solution makes the denominator go to zero then you throw out that solution 5 Radical Equations Isolate the radical expression Ifthere are two radical expressions isolate one ofthem either one Then raise both sides of the equation to the powerthat will undo the radical So if it39s a square root then you would square both sids If it39s a cube root then cube both sides Ifthere is still a radical expression left in the equation repeat this process until there are no radical expressions left Then the equation will either be a linear or quadratic equation and you can solve appropriately v alm v 32vm2 foi63v 3x9 x763 V39xi6 3 V 9x9 x7696yquotx76x9 Example x36v x76x9 Note lfyour original equation has an even root in it then you MUST check your solution by plugging it into the original equation to see if it really works If it doesn39t work then throw it out 6 Exponential Equations Isolate the exponential expression on one side of the equation Then rewrite the equation as its corresponding logarithmic equation using the de nition of Logarithm Then the equation will either be a linear or quadratic equation and you can solve appropriately Example 27382 5 50 400 2738quot 5 50 2736quot 5 400273e2quot 5 50400273e2quot 5 7 Logarithmic Equations Isolate the logarithmic expression on one side ofthe equation Ifthere are several logarithmicterms you39ll need to use logarithmic properties to condense them into one logarithmic expression rst Then rewrite the equation as its corresponding exponential form using the de nition of Logarithm The equation will most likely either be a linear or quadratic equation to nish solving Example 210g4x7310g4x716 210g4x710g4x7163 log4x2710g4x7163 2 10g4 x x716 2 3 x x716 43x716x2 64X76416x2 0x2764x1024 0x732x732 x7320 x32 Note For logarithmic equations you MUST check your solution by plugging it into the original equation to see ifit is in the domain of the logarithm We can only take the log of positive numbers so ifour proposed solution makes us try to take the log ofa negative number or zero then we must throw out that solution